Unfortunately, the definition that's missing is the one that seems to be most prevalent these days. It's an extension of Nr. 8 and means "Lösungsansatz", "Lösungsversuch", "Lösungsmethode" and is used in scientific and political contexts. Often with the verb "verfolgen", e.g. "Wir haben in den letzten Monaten einen neuen Ansatz verfolgt, um dieses Problem zu lösen."

Die Förderalismuskommission hat bislang noch keinen konkreten Reformvorschlag geliefert. [this is the extra "modern" meaning I was talking about - Note, that though "Vorschlag" is a synonym here, it has nothing to do with meaning 7, although it seems to border on it]

Paraphrase: Multikulti ist gescheitert, absolut gescheitert!
...Sorry, this sentence is just a mess - I suspect the author wants to say something like "Versuch" or something. This is a great example for a certain creepy ghostlike quality of some words in public speech. You somehow get the gist, or can kind of "feel" what's supposed to be meant, but can't really nail the meaning down. (Same problem as with "nachhaltig" cringe)

The first three examples can be rephrased with "(gewählte/begonnene) Herangehensweise". Here "Ansatz" is used like in "zu einem Sprung ansetzen" or "zu einem Schlag ansetzen" and describes the starting/preparation of a process. It emphasises the fact that the chosen approach has just begun and is far from completed.

With example 4 I am with Takkat ("in Ansätzen" = "andeutungsweise"), again emphasizing the state of non-completion.